{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

16bf10lec9

# 16bf10lec9 - Math 16b Thomas Scanlon Autumn 2010 Thomas...

This preview shows pages 1–9. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Math 16b Thomas Scanlon Autumn 2010 Thomas Scanlon Math 16b Chain rule recalled (f ยท g ) (x ) = f (x )g (x ) + f (x )g (x ) Thomas Scanlon Math 16b Inverting the Chain Rule: Integration by Substitution (f ยท g ) (x ) = f (x )g (x ) + f (x )g (x ) f (x )g (x ) = (f ยท g ) (x ) โ f (x )g (x ) f (x )g (x )dx = (f ยท g ) (x )dx โ g (x )f (x )dx g (x )f (x )dx f (x )g (x )dx = f (x )g (x ) โ Thomas Scanlon Math 16b Formalism of Integration by Parts Often, one nds two functions u and v so that the integrand may be written as udv where dv = v (x )dx . If we succeed in so doing, then from the equality u (x )v (x ) = u (x )v (x ) โ v (x )u (x ) we see that if f (x ) = u (x )v (x ), then f (x )dx = u (x )v (x )dx = udv = uv โ vdu If vdu is easier to evaluate than is succeeds. Thomas Scanlon Math 16b f (x )dx , then the method An integral revisited Take u = x and v = โ cos(x ) so that dv = sin(x )dx and du = dx . Then, x sin(x )dx = = u dv uv โ v du cos(x )dx = โx cos(x ) + = โx cos(x ) + sin(x ) + C Thomas Scanlon Math 16b Example Integrate: xe x dx Take u = x and v = e x so that du = dx and dv = e x dx . x e x dx = = = = u dv uv โ xe x โ v du e x dx xe x โ e x + C Thomas Scanlon Math 16b Another example Integrate e x cos(x )dx Set u = e x and v = sin(x ) so that du = e x dx and dv = cos(x )dx . Then e x cos(x )dx = = = = u dv uv โ v du e x sin(x )dx e x sin(x )dx e x sin(x ) โ e x sin(x ) โ Thomas Scanlon Math 16b Solution, continued e x cos(x )dx = e x sin(x ) โ e x sin(x )dx e x cos(x )dx e x cos(x )dx e x cos(x )dx = e x sin(x ) + e x cos(x ) โ e x cos(x )dx = e x sin(x ) + e x cos(x ) โ 2 e x cos(x )dx = e x sin(x ) + e x cos(x ) Thomas Scanlon Math 16b 1 1 2 2 x and y = cos(x ) so that dw = e x dx and Set w = e dy = โ sin(x )dx Then x e x cos(x )dx = e x sin(x ) + e x cos(x ) + C A third example Integrate ln(x )dx Set u = ln(x ) and v = x so that du = ln(x )dx = = = = = Thomas Scanlon Math 16b dx x and dv = dx . u dv uv โ v du x dx x dx 1 x ln(x ) โ x ln(x ) โ x ln(x ) โ x + C ...
View Full Document

{[ snackBarMessage ]}

### Page1 / 9

16bf10lec9 - Math 16b Thomas Scanlon Autumn 2010 Thomas...

This preview shows document pages 1 - 9. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online